Although Split seems further to Korcula than Dubrovnik, due to the introduction of cheap flights to Split a couple of years ago especially Easy Jet’s flights to Split people often fly there, instead of Dubrovnik Airport which is actually Korcula Town’s nearest airport.
Another reason why Split is a good access point to Korcula is that ferries from Ancona and Pescara (Italy) arrive here so people who arrived by early morning ferry from Ancona to Split can get to Korcula on the same day. Also, if you are visiting Vela Luka, getting there from Split is a better (faster) option.
The only car ferry that goes from Split to Korcula island is the one that sails to Vela Luka. There are no car ferries from Split to Korcula town. If you miss that one, you have to drive along the coast and get to the island via Orebic to Korcula ferry. (see map). If you decide or have to drive along the coast, consider taking a car ferry from Ploce to Trpanj which will shorten your driving time and give you an opportunity to enjoy this one-hour long sail that offers some beautiful views over the channel.
The buses and ferries are in operation every day so they are pretty reliable means of transport. They are comfortable and air-conditioned and there are some nice views of the islands and bays as you travel along the coast of the Adriatic Sea.
It is also worth mentioning that sometimes, if its very very bad weather, some of the fast ferries are not in service so I would suggest you, in such case, take the slow car ferry to Vela Luka as this one is slow but sturdy and capable to cope with big waves.
All ferries are run by Jadrolinija or Krilo Kapetan Luka ferry companies. Some of them offer advance online booking while others are booked on a first-come, first-served basis. Check jadrolinija.hr and krilo.hr website to see what they offer for advance booking right now. Also always consult croatiaferries.com site where you can find updates for all ferries in Croatia.
Update: see articles I wrote about getting from Brac island to Dubrovnik and from Hvar to Dubrovnik which is both relevant to getting to Korcula (and this one for the opposite direction) Some info in these articles may be useful for your travel options as ferries also stop in Korcula.